Guaranteed Bounds for Solution ofParameter Dependent System of Equations

Andrew Pownuk1, Iwona Skalna2, and Jazmin Quezada 1

1 - The University of Texas at El Paso, El Paso, Texas, USA2 - AGH University of Science and Technology, Krakow, Poland

22th Joint UTEP/NMSU Workshop on Mathematics,Computer Science, and Computational Sciences

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Outline

1 Solution Set

2 Interval Methods

3 Optimization methods

4 New Approach

5 Example 1

6 Example 2

7 Example 3

8 Conclusions

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Solution of PDE

Parameter dependent Boundary Value Problem

A(p)u = f (p), u ∈ V (p), p ∈ P

Exact solution

u = infp∈P

u(p), u = supp∈P

u(p)

u(x , p) ∈ [u(x), u(x)]

Approximate solution

uh = infp∈P

uh(p), uh = supp∈P

uh(p)

uh(x , p) ∈ [uh(x), uh(x)]

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Mathematical Models in Engineering

Linear and nonlinear equations.Multiphysics (solid mechanics, fluid mechanics etc.)Ordinary and partial differential equations, variationalequations, variational inequalities, numerical methods,programming, visualizations, parallel computing etc.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Two point boundary value problem

Sample problem− (g(x , p)u′(x))′ = f (x , p)

u(0) = 0, u(1) = 0

and uh(x) is finite element approximation given by a weakformulation

1∫0

g(x , p)u′h(x)v ′(x)dx =

1∫0

f (x , p)v(x)dx ,∀v ∈ V(0)h

ora(uh, v) = l(v), ∀v ∈ V

(0)h ⊂ H1

0

where uh(x) =n∑

i=1uiϕi (x) and ϕi (xj) = δij .

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

The Finite Element Method

Approximate solution

1∫0

g(x , p)u′h(x)v ′(x)dx =

1∫0

f (x , p)v(x)dx

.

n∑j=1

n∑i=1

1∫0

g(x , p)ϕi (x)ϕj(x)dxui −1∫

0

f (x , p)ϕj(x)dx

vj = 0

Final system of equations (for one element) Ku = q where

Ki ,j =

1∫0

g(x , p)ϕi (x)ϕj(x)dx , qi =

1∫0

f (x , p)ϕi (x)dx

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Global Stiffness Matrix

Global stiffness matrix

n∑p=1

n∑q=1

ne∑e=1

neu∑i=1

neu∑j=1

Uej ,p

∫Ωe

g(x , p)∂ϕe

i (x)

∂x

∂ϕej (x)

∂xdxUe

i ,quq−

n∑q=1

ne∑e=1

neu∑i=1

neu∑j=1

Uej ,p

∫Ωe

f (x , p)ϕei (x)ϕe

j (x)dx

vp = 0

Final system of equations

K (p)u = q(p)⇒ F (u, p) = 0

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Solution Set

Nonlinear equation F (u, p) = 0 for p ∈ P.

F : Rn × Rm → Rn

Implicit function u = u(p)⇔ F (u, p) = 0

u(P) = u : F (u, p) = 0, p ∈ P

Interval solution

ui = minu : F (u, p) = 0, p ∈ P

ui = maxu : F (u, p) = 0, p ∈ P

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Interval Methods

A. Neumaier, Interval Methods for Systems of Equations(Encyclopedia of Mathematics and its Applications, CambridgeUniversity Press, 1991.

Z. Kulpa, A. Pownuk, and I. Skalna, Analysis of linearmechanical structures with uncertainties by means of intervalmethods, Computer Assisted Mechanics and EngineeringSciences, 5, 443-477, 1998.

V. Kreinovich, A.V.Lakeyev, and S.I. Noskov. Optimal solutionof interval linear systems is intractable (NP-hard). IntervalComputations, 1993, 1, 6-14.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Interval Methods

T. Burczynski, J. Skrzypczyk, Fuzzy aspects of the boundaryelement method, Engineering Analysis with BoundaryElements, Vol.19, No.3, pp. 209216, 1997

A. Neumaier and A. Pownuk, Linear Systems with LargeUncertainties, with Applications to Truss Structures, Journal ofReliable Computing, 13(2), 149-172, 2007.

Muhanna, R. L. and R. L. Mullen. Uncertainty in MechanicsProblemsInterval-Based Approach, Journal of EngineeringMechanics 127(6), 557-566, 2001.

I. Skalna, A method for outer interval solution of systems oflinear equations depending linearly on interval parameters,Reliable Computing, 12, 2, 107-120, 2006.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Optimization methods

Interval solution

ui = minu(p) : p ∈ P = minu : F (u, p) = 0, p ∈ P

ui = maxu(p) : p ∈ P = maxu : F (u, p) = 0, p ∈ P

ui =

min uiF (u, p) = 0p ∈ P

, ui =

max uiF (u, p) = 0p ∈ P

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

KKT Conditions

Nonlinear optimization problem for f (x) = ximinx

f (x)

h(x) = 0g(x) ≥ 0

Lagrange function L(x , λ, µ) = f (x) + λTh(x)− µTg(x)Optimality conditions can be solved by the Newton method.

∇xL = 0∇λL = 0µi ≥ 0

µigi (x) = 0h(x) = 0g(x) ≥ 0

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

KKT Conditions - Newton Step

F ′(X )∆X = −F (X )

F ′(X ) =

(∇2

x f (x) +∇2xh(x)y

)n×n ∇xh(x)n×m −In×n

(∇xh(x))Tm×n 0n×m 0m×nZn×n 0n×m Xm×n

∆X =

∆x∆y∆z

,X =

xyz

F (X ) = −

∇x f (x) +∇xhT (x)y − z

h(x)XYe − µke

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Steepest Descent Method

In order to find maximum/minimum of the function u it ispossible to apply the steepest descent algorithm.

1 Given x0, set k = 0.

2 dk = −∇f (xk). If dk = 0 then stop.

3 Solve minαf (xk + αdk) for the step size αk . If we know

second derivative H then αk =dTk dk

dTk H(xk )dk

.

4 Set xk+1 = xk + αkdk , update k = k + 1. Go to step 1.

I. Skalna and A. Pownuk, Global optimization method forcomputing interval hull solution for parametric linear systems,International Journal of Reliability and Safety, 3, 1/2/3,235-245, 2009.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

New Approach for Finding Guaranteed Bounds

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

New Approach for Finding Guaranteed Bounds

Theorem

Let’s assume that g : P → R is a continuous function, P is apath-connected, compact subset of R, theng(P) = g(p) : p ∈ P = [g(pmin), g(pmax)] = [xmin, xmax ] is aclosed interval and pmin, pmax ∈ P,xmin = infg(x) : p ∈ P, xmax = supg(x) : p ∈ P.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

New Approach for Finding Guaranteed Bounds

Theorem

Let’s assume that g : P → R is a continuous function, P is apath-connected, compact subset of Rm, we know at least onevalue x0 = g(p0) such that p0 ∈ P, and exists some ε > 0 suchthat x0 + ∆x /∈ g(P) for all ∆x ∈ (0, ε], thenx0 = g(pmax) = gmax is a guaranteed upper bound of the setg(P).

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

New Approach for Finding Guaranteed Bounds

Theorem

Let’s assume that g : P → Rn is a continuous function that isdefined as a unique solution of the equation f (x , p) = 0, P is apath-connected, compact subset of Rm, we know at least onevalue x0 = g(p0) such that p0 ∈ P, and exists some ε > 0 suchthat x0,i + ∆xi /∈ gi (P) for all ∆xi ∈ (0, ε], thenx0,i = gi (pmax) = gi ,max is a guaranteed upper bound of the setgi (P) = [gi ,min(P), gi ,max(P)].

If the equation f (x , p) = 0 has multiple solutions x = gi (p)(i = 1, ..., s), then

g(P) = g1(P) ∪ g2(P) ∪ ... ∪ gs(P)

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 1

Let’s consider the equation nonlinear equation with uncertainparameter

x2 − 4p2 = 0 for p ∈ [1, 2].

Presented equation has two solutions x = g1(p) = 2p andx = g2(p) = −2p.Non-guaranteed solutions are[x1, x1] = g1([1, 2]) = [2, 4] and[x2, x2] = g2([1, 2]) = [−4,−2].

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 1

Let’s check if the number x1 = 4 + ∆x is a solution for ∆x > 0.

0 ∈ f (4 + ∆x , [1, 2])

0 ∈ (4 + ∆x)2 − 4[1, 2]2

0 ∈ 16 + 8∆x + ∆x2 − 4[1, 4]

0 ∈ 16 + 8∆x + ∆x2 − [4, 16]

0 ∈ [8∆x + ∆x2, 12 + 8∆x + ∆x2]

Last condition is not satisfied then x1 = 4 + ∆x is not asolution for any ∆x > 0 then x = 4.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 1

Let’s check if the number x1 = 2−∆x is a solution for ∆x > 0.

0 ∈ f (2−∆x , [1, 2])

0 ∈ (2−∆x)2 − 4[1, 2]2

0 ∈ 4− 4∆x + ∆x2 − 4[1, 2]

0 ∈ 4− 4∆x + ∆x2 + [−8,−4]

0 ∈ [−4− 4∆x + ∆x2,−4∆x + ∆x2]

For small ∆x it is possible to neglect the quadratic term and−4∆x + ∆x2<0. Last condition is not satisfied thenx1 = 2−∆x is not a solution for small ∆x > 0 then x = 2.The interval solution [x1, x1] = [2, 4] is guaranteed.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 2

Let’s consider the following system of linear interval equations[−4,−3] x1 + [−2, 2] x2 = [−8, 8][−2, 2] x1 + [−4,−3] x2 = [−8, 8]

(1)

Let’s assume that the non-guaranteed solution isx1 ∈ [−8, 8], x2 ∈ [−8, 8].

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 2

Let x1 = 8 + ∆x1 and ∆x1 > 0.a11(8 + ∆x1) + a12x2 = b1

a21(8 + ∆x1) + a22x2 = b2a11(8 + ∆x1) + a12

b2−a21(8+∆x1)a22

= b1

x2 = b2−a21(8+∆x1)a22

a11(8 + ∆x1) +a12b2

a22− a12a21(8 + ∆x1)

a22= b1

(8 + ∆x1)

(a11 −

a12a21

a22

)+

a12b2

a22− b1 = 0

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 2

(8 + ∆x1)

(a11 −

a12a21

a22

)+

a12b2

a22− b1 = 0

0 ∈ (8 + ∆x1)

(a11 −

a12a21

a22

)+

a12b2

a22− b1

0 ∈ (8 + ∆x1)

[−16

3,−5

3

]+

[−16

3,

16

3

]− [−8, 8]

0 ∈[−128

3− 16

3∆x1,−

40

3− 5

3∆x1

]+

[−40

3,

40

3

]0 ∈

[−56− 16

3∆x1,−

5

3∆x1

]Then x1 = 8 + ∆x1 is not a solution for ∆x1 > 0 and x1 = 8.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Parametric Linear System of Equations

Sample boundary value problem

d

dx

(EA

du

dx

)= 0, u(0) = 0,EA

du(L)

dx= P

After discretizationk1 + k2 −k2 0 ... 0 0−k2 k2 + k3 −k3 ... 0 0

...0 0 0 ... kn−1 + kn −kn0 0 0 ... −kn kn

u1

u2

...un−1

un

=

00...0P

Let n = 2 and k1, k2 ∈

[13 ,

12

]and P = 1 then the

non-guaranteed solution is u1 ∈ [2, 3], u2 ∈ [4, 6].

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 3

Let’s assume that u1 = 3 + ∆u1, ∆u1 > 0 then(k1 + k2)(3 + ∆u1)− k2u2 = 0−k2(3 + ∆u1) + k2u2 = P(k1 + k2)(3 + ∆u1)− k2u2 = 0

u2 = P+k2(3+∆u1)k2

(k1 + k2)(3 + ∆u1)− k2P + k2(3 + ∆u1)

k2= 0

(k1 + k2)(3 + ∆u1)− P − k2(3 + ∆u1) = 0

k1(3 + ∆u1)− P = 0

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Example 3

By assumption k1 ∈[

13 ,

12

], P = 1 then

0 ∈[

1

3,

1

2

](3 + ∆u1)− P,

0 ∈[

1

3(3 + ∆u1),

1

2(3 + ∆u1)

]− 1,

0 ∈[

1

3∆u1,

1

2+

1

2∆u1

].

The last condition cannot be satisfied because ∆u1 > 0consequently u1 = 3 + ∆u1 cannot be a solution of the systemfor any ∆u1 > 0 and 3 = u1 is guaranteed upper-bound of thesolution u1.

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Solution Set

IntervalMethods

Optimizationmethods

New Approach

Example 1

Example 2

Example 3

Conclusions

Conclusions

Methodology presented in this paper can be applied for widerange of parameter dependent system of equations andeigenvalue problems.The method can be applied not only for the solution of theequations with set-valued parameters but also for finding valuesof the functions that depends of such solutions which is veryimportant in the practical applications.By using theory from the presentation, in some cases, in orderto use guaranteed bounds of the solution it is possible existing,well established computational methods and at the end provethat the solution is guaranteed.Methodology presented in this presentation can be applied toselected solutions or to all solutions of the systems of nonlinearequations.

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